Chromatic bounds for some classes of 2K2-free graphs

Abstract A hereditary class G of graphs is χ - bounded if there is a χ - binding function , say f such that χ ( G ) ≤ f ( ω ( G ) ) , for every G ∈ G , where χ ( G ) ( ω ( G ) ) denotes the chromatic (clique) number of G . It is known that for every 2 K 2 -free graph G , χ ( G ) ≤ ω ( G ) + 1 2 , and the class of ( 2 K 2 , 3 K 1 )-free graphs does not admit a linear χ -binding function. In this paper, we are interested in classes of 2 K 2 -free graphs that admit a linear χ -binding function. We show that the class of ( 2 K 2 , H )-free graphs, where H ∈ { K 1 + P 4 , K 1 + C 4 , P 2 ∪ P 3 ¯ , H V N , K 5 − e , K 5 } admits a linear χ -binding function. Also, we show that some superclasses of 2 K 2 -free graphs are χ -bounded.